Croatica Chemica Acta, Vol. 69 No. 3, 1996.
Original scientific paper
Combinatorial Self-Similarity
Sherif El-Basil
; Faculty of Pharmacy, Kasr El-Ainist., Cairo, 11562 Egypt
Abstract
Combinatorial (or numerical) self-similarity is an apparently new
concept, introduced here in an attempt to explain the similarity of
properties of the members of a homologous series that are not (geometrically) self-similar and whence are not (deterministic) fractals. The term is defined in the following steps:
a) Select a numerical invariant, ep, characteristic of the member
of the series
b) Partition this property, ep, into a finite number of parts
through a prescribed algorithm
c) Members are described so as to be combinatorially self-similar
(or to represent a »numerical« fractal) if the limits of the ratios
of ep of two successive members at infinite stages of homologation
are equal for all parts, and equal the corresponding limit for the
total property. In the present work, ep is taken to be the Kekule
count, K, when dealing with benzenoid systems and the topological
index, Z, (H. Hosoya, Buzz. Chem. Soe. Japan 44 (1971) 2332) when
dealing with saturated hydrocarbons. The previously described
equivalence relation, l, [S. El-Basil, J. Chem. Soe. Faraday Trans. 89
(1993) 909; J. Mol. Struet. (Theochem) 288 (1993) 67; J. Math. Chem.
14 (1993) 305; J. Mol. Struet. (Theoehem) 313 (1994) 237; J. Chem.
Soe. Faraday Trans. 90 (1994) 2201], is used to partition K when the
number of terminal hexagons remains constant throughout the series;
otherwise the method of Klein and Seitz [D. J. Klein and W.A. Seitz,
J. Mol. Struet. (Theochem) 169 (1988) 167] is used. For alkanes, an
appropriate recurrence relation is used to partition the Z values. It
was found that ep for any homologous series of unbranehed benzenoids, alkanes, Clar graphs, rook broads and King polyominos are all scaled by the golden mean, t = 1,618033989, while homologous series of other types of benzenoids also represent »numerical«fractals, but the characteristic scaling factors depend on the closed form expressions of their K values. In all cases, self-similarities were manifested by expressing the ratios of adjacent t.p's in the form of continued fractions, which in some cases led to exact selfsimilarity but in most cases self-similarity was only approximate.
Keywords
Hrčak ID:
177134
URI
Publication date:
1.11.1996.
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